Dirichlet and Neumann eigenvalue problems on CR manifolds. (English) Zbl 1405.32057
The main results in this paper are solutions to the Dirichlet and Neumann problems for the sub-Laplacian on domains in CR manifolds.
Let \(M\) be a strictly pseudoconvex CR manifold, \(\theta\) be a positively oriented contact form on \(M\) and \(d_{\theta}\) be a Carnot-Carathéodory metric on \((M,\theta)\). The authors prove that the weak Dirichlet problem for the sub-Laplacian can be solved on domains supporting the Poincaré inequality, and the weak Neumann problem can be solved on \(C^{1,1}\) connected \((\epsilon, \delta)\) domains in the Heisenberg group. As an application the authors show the discretness of the Dirichlet and Neumann spectra for the sub-Laplacian on bounded domains of Carnot-Carathéodory complete pseudohermitian manifolds.
Let \(M\) be a strictly pseudoconvex CR manifold, \(\theta\) be a positively oriented contact form on \(M\) and \(d_{\theta}\) be a Carnot-Carathéodory metric on \((M,\theta)\). The authors prove that the weak Dirichlet problem for the sub-Laplacian can be solved on domains supporting the Poincaré inequality, and the weak Neumann problem can be solved on \(C^{1,1}\) connected \((\epsilon, \delta)\) domains in the Heisenberg group. As an application the authors show the discretness of the Dirichlet and Neumann spectra for the sub-Laplacian on bounded domains of Carnot-Carathéodory complete pseudohermitian manifolds.
Reviewer: Koichi Saka (Akita)
Keywords:
CR manifolds; Poincaré inequality; Carnot-Carathéodory metric; sub-Laplacian; Dirichlet and Neumann problemsReferences:
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